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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"299ef05bac7ecf52c0d5ed60cf83230f0b959f08ba7829fb8e26c3d4686a6505","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cd3918419c5686a8","role":"sequence","name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,      1;\n   3:      2,      4,       1;\n   4:      5,     15,       8,      1;\n   5:     16,     62,      50,     13,      1;\n   6:     65,    297,     311,    121,     19,     1;\n   7:    326,   1664,    2079,   1050,    245,    26,     1;\n   8:   1957,  10815,   15331,   9294,   2815,   442,    34,    1;\n   9:  13700,  80450,  125418,  87017,  31759,  6475,   735,   43,  1;\n  10: 109601, 675745, 1135270, 874194, 367227, 90475, 13342, 1150, 53, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-9*n+20)*P(n-4,x) + (2*n^2-13*n+20+(2*n-8)*x)*P(n-3,x) - (n^2-2*n-3+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 2 + 4*x + x^2, P(4,x) = 5 + 15*x + 8*x^2 + x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 2 + 4*x + x^2;\nP[4, x_] := 5 + 15*x + 8*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 9*n + 20)*P[n - 4, x] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*P[n - 3, x] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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